What is Mathematical Epidemiology?

What is mathematical epidemiology? Well, mathematical epidemiology is when mathematicians use math to predict outcomes in various statistical problems. These problems include growth in infectious bacteria, change in population, and even the effects of climate change. Why is this used? It is used because it doesn’t need a complete set of data to figure out a solution, as long as you can create an equation and plug in the values.

Who uses it? Mathematicians and scientists use it in fields such as biotechnology, medical science, civil engineering, and as public health professionals.

Hello! And welcome to the realm of quantum mechanics! First off, what in the world is quantum mechanics? Let’s start with a brief introduction.

What is Quantum Mechanics?

Quantum mechanics is one of the most important branches of physics. It focuses on the laws of nature at three different levels: molecular, atomic, and subatomic. Quantum mechanics has a variety of important concepts; the following are some that we learned through our problem set: Planck’s law, the photoelectric effect, and wave-particle duality. A crucial element of quantum mechanics is understanding that everything has characteristics of both waves and particles. We will touch on this and many other topics later on.

Intro to Number Systems

As children we grew up counting in the base ten system (1, 2, 3, etc). However, base ten is only one of many numerical systems. Over these past to weeks at Girls Talk Math at UNC, our task was to explore other number systems that are not as frequently used as the base 10 system, specifically binary and hexadecimal number systems.

Binary

The exact definition of binary is related to using a system of numerical notation that has 2 rather than 10 as a base. This means only two single digits are used, 0 and 1.

Binary is used for data storage. Binary basically makes it easier for computer processors to understand and interpret incoming information/instructions.

Binary was first discussed by Gottfried Leibniz in 1689 but binary numerical systems were not put to use until a binary converter was created hundreds of years later. The binary system was officially implemented just before the beginning of the nineteenth century.

Data is all around us, but it has to be studied in some way, right? How else are we supposed to know what it’s about? That’s what graph theory and network science are for! To organize and connect data mathematicians use networks and graphs as well as scientific computing (like coding).

Network science is an application-based study of graphs. To understand network science, we first have to understand the graphs:

Graph Theory

Graphs represent data through nodes, which are the separate points of a graph, and edges, which connect the nodes. There are two types of graphs: directed and undirected graphs. Directed graphs rely on the order of the vertices to be the same, while undirected graphs don’t rely on the order of the nodes.

Euler Characteristics

Euler characteristics are defined by the equation V- E + F = 2 where V = number of vertices, E = number of edges or nodes, and F = number of faces. Sometimes though, the equation V – E + F = 2 does not work for all situations because the solution can give various outcomes due to the dimensions and simplicity of the object. If 2 objects are topologically the same, they will have the same Euler characteristics. For all simple polygons, the Euler characteristics equal one. Figures with holes don’t follow these conventions as the holes in these figures add additional faces and edges not proportional to the formulas for simple figures.

*Brooke helped the group work through the problem set but was unfortunately unable to attend camp during the blog writing.

What We Did:

Knot theory has many different applications in math including algebra and geometry, and (outside of math) physics. We learned that we can use algebraic techniques to describe knots. When trying to understand knot theory we learned that it is very helpful to work in a group and read the definitions out loud. Us working together was key in understanding knot theory.

RSA was one of the first public-key* cryptosystems and it is widely used for secure data transmission. It was first created by Ron Rivest, Adi Shamir, and Leona Adleman.

*Public key is used to establish a secret key, and the public key is sent in public. We then use the private key method to encrypt and decrypt large amounts of data, but no one knows the private key.

To code: U^s=x X(mod N)=Y

To decode Y^t=O O(mod N)=U

In computing, the modulo operation finds the remainder after division of one number by another. Given two positive numbers, a and n, a modulo n (in other words a mod n) is the remainder of the a division of a by n, where a is the dividend and n is the divisor.

*Alana and Tamarr helped the group work through the problem set but were unfortunately unable to attend camp during the blog writing.

For years, people have been trying to find a way to send secret messages. This may have been easy to do in the ancient times of the Roman Empire, where you could write a message, and then hand-deliver it to your recipient. This way, you could be certain that nobody else could intercept it. However, this becomes a lot more difficult in today’s online tech-driven world. People no longer hand-deliver letters; rather, we email or text our friends. So how do we make sure that nobody else can intercept your text message as it travels the internet before finally landing on your friend’s cell-phone? The answer is found in cryptography, a technology that is becoming more and more important in today’s world. Today, we are going to focus on one particular form of cryptography: elliptic curve cryptography.

*Nevaeh helped the group work through the problem set but was unfortunately unable to attend camp during the blog writing.

Mathematical Epidemiology explores the realm of mathematics applied to public health. It relies on modeling to use known information about certain scenarios regarding the spread of diseases and then uses it to predict future outcomes. By the end of the problem set, our group learned about the challenging process that comes with trying to predict population sizes in order to control the spreading of diseases. The equations that are faced in this branch of mathematics are at the heart of mathematical modeling.

Mathematical Models and Modeling

A mathematical model is an equation used to predict or model the most likely results to occur in a real-world situation. We used these types of equations to model the spread of a disease in a population, tracking the flow of populations from susceptible to infected to recovered. In real life scenarios, there are too many variables to fully account for, so we only were able to place a few in our equations. This made the models less accurate, but at the same time very useful to us in our problem set. They gave us a good idea of how things worked in an actual epidemic and helped us to understand what mathematical modeling really is.

Quantum Mechanics is the physics of molecular and microscopic particles. However, it has applications in everyday life as well. If someone asked you if a human was a particle or a wave, what would you think? What about a ball? What about light? Not so easy now, is it? It turns out that all of those things, and in fact, everything around us, can be expressed in physics as both a particle and a wave. This might seem a little unbelievable, but for now, let’s start with the basics.

Classical Physics

Although Classical Physics sounds like a complicated idea, it’s the most simple branch of physics. It’s what you think of when someone says “physics”. Classical Physics lays the basic foundation to Quantum Physics with a few basic laws.